\section{Introduction}
This is the introduction \ldots
\subsection{$TM_z$ Basis Field Expansion}
From (\ref{eqn:sepfcnsTM}) the magnetic vector potential is defined as
\begin{align}
    A_z(\rho,\phi,z)&=\sum_{m,n}(C_{A})_{m,n}a_{z}(\rho,\phi)_{m,n}\frac{i_A(z)_{m,n}}{\sqrt{(Z_A)_{m,n}}}\label{eqn:sepfcnsTMzcyl}
\end{align}
Using seperation of variables in cylindrical coordinates
\begin{align}
    a_{z}(\rho,\phi)_{m,n} &= R_A(\rho)_{m,n}\Phi_A(\phi)_{m}	
\end{align}
where each function and its respective derivative is given as
\begin{align}
	R_A(\rho)_{m,n} &= \mathrm{J}_m\bigl((k_{A\rho})_{m,n}\;\rho\bigr)\\
	R_A'(\rho)_{m,n} &= -(k_{A\rho})_{m,n}\mathrm{J}_{m+1}\bigl((k_{A\rho})_{m,n}\;\rho\bigr)+\frac{m}{\rho}\mathrm{J}_{m}\bigl((k_{A\rho})_{m,n}\;\rho\bigr)
\end{align}
where $'=\frac{\partial}{\partial \rho}$ and 
\begin{align}
	\Phi_A(\phi)_{m} &= \cos(m\phi)\\
	\Phi_A'(\phi)_{m} &= -m\sin(m\phi)
\end{align}
where $'=\frac{\partial}{\partial \phi}$. From the \emph{contraint equation} it is found that
\begin{align}
	(k_{Az})_{m,n} = -j\sqrt{(k_{A\rho})_{m,n}^2-k^2}
\end{align}
where 
\begin{align}
	k &= \omega\sqrt{\varepsilon'\mu'}\\
	(k_{A\rho})_{m,n} &= \frac{\chi_{m,n}}{a}
\end{align}
in which $a$ is the radius of the circular waveguide and $\chi_{m,n}$ is the $n^{th}$ zero of the Bessel function $\mathrm{J}_m(\rho)$.
From (\ref{eqn:ZA}) a $TM_z$ wave impedance is defined as
\begin{align}
	(Z_A)_{m,n}&=\frac{(k_{Az})_{m,n}}{\omega\varepsilon'}\label{eqn:ZAcyl_2} 
\end{align}
The normalized current $i(z)$ is expanded into traveling waves as,
\begin{align}
\begin{split}
    i(z)_{m,n}&=(i^+)_{m,n}e^{-j(k_{Az})_{m,n}z}-(i^-)_{m,n}e^{j(k_{Az})_{m,n}z}\\
    &=b_A(z)_{m,n}-a_A(z)_{m,n}\label{eqn:solnscalarwavenormTMcyk}
\end{split}
\end{align}
and whos derivative is,
\begin{align}
\begin{split}
    \frac{di(z)_{m,n}}{dz}&=-j \bigl(k_{Az})_{m,n}((i^+)_{m,n}e^{-j(k_{Az})_{m,n}z}+(i^-)_{m,n}e^{j(k_{Az})_{m,n}z}\bigr)\\
    &=-j (k_{Az})_{m,n}\bigr(b_A(z)_{m,n}+a_A(z)_{m,n}\bigl)\label{eqn:solnscalarwavenormderivTMcyl}
\end{split}
\end{align}
where $a(z)$ is defined as a wave that is incident on the port and $b(z)$ is defined as a wave reflected from the port. 
Expanding the transverse gradient operator in cylindrical coordinates 
\begin{align}
\vec{\nabla}_\bot = \frac{\partial}{\partial\rho}\hat{\rho}+\frac{1}{\rho}\frac{\partial}{\partial\phi}\hat{\phi}
\end{align}
the basis electric fields are obtained from (\ref{eqn:EABasis})
\begin{align}
    \vec{e}_A(\rho,\phi)_{m,n}&=-(C_A)_{m,n}\sqrt{(Z_A)_{m,n}}\vec{\nabla}_\bot a_z(\rho,\phi)_{m,n}\label{eqn:EABasisCyl}\\
    &=-(C_A)_{m,n}\sqrt{(Z_A)_{m,n}}\left(R'(\rho)_{m,n}\Phi(\phi)_{m}\hat{\rho}+\frac{1}{\rho}R(\rho)_{m,n}\Phi'(\phi)_{m}\hat{\phi}\right)
\end{align}
and the magnetic basis fields are given from (\ref{eqn:HABasis}) to get
\begin{align}
    \vec{h}_A(\rho,\phi)_{m,n}&=\frac{(C_A)_{m,n}}{\sqrt{(Z_A)_{m,n}}}\vec{\nabla}_\bot a_z(\rho,\phi)_{m,n}\times\hat{z}\label{eqn:HABasisCyl}\\
    &=\frac{(C_A)_{m,n}}{\sqrt{(Z_A)_{m,n}}}\left(\frac{1}{\rho}R(\rho)_{m,n}\Phi'(\phi)_{m}\hat{\rho}-R'(\rho)_{m,n}\Phi(\phi)_{m}\hat{\phi}\right)
\end{align}
The normalization constant is found from (\ref{eqn:normconstA}) to be
\begin{align}
	(C_A)_{m,n} &= \left[\frac{-\pi}{2}(\chi_{m,n})^2 \mathrm{J}_{m-1}(\chi_{m,n})\mathrm{J}_{m+1}(\chi_{m,n})(1+\delta_{m,0})\right]^{-\frac{1}{2}}\label{eqn:normconstAcyl_2}	
\end{align}
The complex power of each mode is found from (\ref{eqn:PowerTM}) to get
\begin{align}
	(P_A)_{m,n} &= \frac{1}{2}\frac{(Z_A)_{m,n}}{|(Z_A)_{m,n}|}[(b_A)_{m,n}+(a_A)_{m,n}][(b_A)_{m,n}-(a_A)_{m,n}]^*\label{eqn:PowerTMcyl}
\end{align}

\subsection{$TE_z$ Basis Field Expansion}
From (\ref{eqn:sepfcnsTE}) the magnetic vector potential is defined as
\begin{align}
    F_z(\rho,\phi,z)&=\sum_{m,n}(C_{F})_{m,n}f_z(\rho,\phi)_{m,n}{\sqrt{(Z_F)_{m,n}}}{v(z)_{m,n}}\label{eqn:sepfcnsTEzcyl}
\end{align}
Using seperation of variables in cylindrical coordinates
\begin{align}
    f_z(\rho,\phi)_{m,n} &= R_F(\rho)_{m,n}\Phi_F(\phi)_{m}	
\end{align}
where each function and its respective derivative is given as
\begin{align}
	R_F(\rho)_{m,n} &= \mathrm{J}_{m}\bigl((k_{F\rho})_{m,n}\;\rho\bigr)\\
	R_F'(\rho)_{m,n} &= -(k_{F\rho})_{m,n}\mathrm{J}_{m+1}\bigl((k_{F\rho})_{m,n}\;\rho\bigr)+\frac{m}{\rho}\mathrm{J}_{m}\bigl((k_{F\rho})_{m,n}\;\rho\bigr)
\end{align}
where $'=\frac{\partial}{\partial \rho}$ and 
\begin{align}
	\Phi(\phi)_{m} &= \sin(m\phi)\\
	\Phi'(\phi)_{m} &= m\cos(m\phi)
\end{align}
where $'=\frac{\partial}{\partial \phi}$. From the \emph{contraint equation} it is found that
\begin{align}
	(k_{Fz})_{m,n} = -j\sqrt{(k_{F\rho})_{m,n}^2-k^2}
\end{align}
where 
\begin{align}
	k &= \omega\sqrt{\varepsilon'\mu'}\\
	(k_{F\rho})_{m,n} &= \frac{\chi_{m,n}'}{a}
\end{align}
in which $a$ is the radius of the circular waveguide and $\chi_{m,n}'$ is the $n^{th}$ zero of the derivative of the Bessel function $\frac{d}{d\rho}\mathrm{J}_m(\rho)$.
From (\ref{eqn:ZF}) a $TE_z$ wave impedance is defined as
\begin{align}
	(Z_F)_{m,n}&=\frac{(k_{Fz})_{m,n}}{\omega\varepsilon'}\label{eqn:ZFcyl_2} 
\end{align}
The normalized current $v(z)$ is expanded into traveling waves as,
\begin{align}
\begin{split}
    v(z)_{m,n}&=(v^+)_{m,n}e^{-j(k_z)_{m,n}z}+(v^-)_{m,n}e^{j(k_z)_{m,n}z}\\
    &=b(z)_{m,n}+a(z)_{m,n}\label{eqn:solnscalarwavenormTEcyl}
\end{split}
\end{align}
and whos derivative is,
\begin{align}
\begin{split}
    \frac{dv(z)_{m,n}}{dz}&=-j (k_z)_{m,n}\bigr((v^+)_{m,n}e^{-j(k_z)_{m,n}z}-(v^-)_{m,n}e^{j(k_z)_{m,n}z}\bigr)\\
    &=-j (k_z)_{m,n}\bigr(b(z)_{m,n}-a(z)_{m,n}\bigl)\label{eqn:solnscalarwavenormderivTEcyl}
\end{split}
\end{align}
where $a(z)$ is defined as a wave that is incident on the port and $b(z)$ is defined as a wave reflected from the port. 
Expanding the transverse gradient operator in cylindrical coordinates 
\begin{align}
\vec{\nabla}_\bot = \frac{\partial}{\partial\rho}\hat{\rho}+\frac{1}{\rho}\frac{\partial}{\partial\phi}\hat{\phi}
\end{align}
the basis electric fields are obtained from (\ref{eqn:EFBasis})
\begin{align}
    \vec{e}_F(\rho,\phi)_{m,n}&={-(C_F)_{m,n}}{\sqrt{(Z_F)_{m,n}}}\vec{\nabla}_\bot f_z(\rho,\phi)_{m,n}\times\hat{z}\label{eqn:EABasisCyl}\\
    &={-(C_F)_{m,n}}{\sqrt{(Z_F)_{m,n}}}\left(\frac{1}{\rho}R(\rho)_{m,n}\Phi'(\phi)_{m}\hat{\rho}-R'(\rho)_{m,n}\Phi(\phi)_{m}\hat{\phi}\right)
\end{align}
and the magnetic basis fields are given from (\ref{eqn:HFBasis}) to get
\begin{align}
    \vec{h}_F(\rho,\phi)_{m,n}&=-\frac{(C_F)_{m,n}}{\sqrt{[Z_F]_{m,n}}}\vec{\nabla}_\bot f_z(\rho,\phi)_{m,n}\label{eqn:HABasisCyl}\\
    &=-\frac{(C_F)_{m,n}}{\sqrt{(Z_F)_{m,n}}}\left(R'(\rho)_{m,n}\Phi(\phi)_{m}\hat{\rho}+\frac{1}{\rho}R(\rho)_{m,n}\Phi'(\phi)_{m}\hat{\phi}\right)
\end{align}
The normalization constant is found from (\ref{eqn:normconstF}) to be
\begin{align}
	(C_F)_{m,n} &= \left[\frac{\pi}{2}(\chi_{m,n}')^2\left\{\mathrm{J}_{m}^2(\chi_{m,n}')-\mathrm{J}_{m-1}(\chi_{m,n}')\mathrm{J}_{m+1}(\chi_{m,n}')\right\}(1-\delta_{m,0})\right]^{-\frac{1}{2}}\label{eqn:normconstFcyl_2}	
\end{align}
The complex power of each mode is found from (\ref{eqn:PowerTE}) to get
\begin{align}
	(P_F)_{m,n} &= \frac{1}{2}\frac{(Z_F)_{m,n}}{|(Z_F)_{m,n}|}[(b_F)_{m,n}+(a_F)_{m,n}][(b_F)_{m,n}-(a_F)_{m,n}]^*\label{eqn:PowerTEcyl}
\end{align}

\section{Modal Radiation}
Assume that $m=1$,
\subsection{Surface Integrals}
\begin{align}
	\iint_{S'}a_{z}(\vec{r_\bot}')_n\;e^{jk\hat{r}\cdot\vec{r}'}\;ds'
	&=\int_0^{a}\rho'\mathrm{J}_1((k_{A\rho})_n\rho')\int_0^{2\pi}\cos(\phi')\;e^{jk\rho'\sin\theta\cos(\phi-\phi')}\;d\phi'\;d\rho'
\end{align}
\begin{align}
	\iint_{S'}f_{z}(\vec{r_\bot}')_i\;e^{jk\hat{r}\cdot\vec{r}'}\;ds'
	&=\int_0^{a}\rho'\mathrm{J}_1((k_{F\rho})_n\rho')\int_0^{2\pi}\sin(\phi')\;e^{jk\rho'\sin\theta\cos(\phi-\phi')}\;d\phi'\;d\rho'
\end{align}

\subsection{Line Integral}
\begin{align}
	\oint_C f_{z}(\vec{r_\bot}')_i e^{jk\hat{r}\cdot\vec{r}'}\hat{u}'\;d\ell'
	&=a\mathrm{J}_1(\chi_{1,n}')\int_0^{2\pi}\sin(\phi')e^{jk\rho'\sin\theta\cos(\phi-\phi')}\hat{\rho}'\;d\phi'\\
	&=a\mathrm{J}_1(\chi_{1,n}')\int_0^{2\pi}\sin(\phi')e^{jk\rho'\sin\theta\cos(\phi-\phi')}(\cos\phi'\hat{x}'+\sin\phi'\hat{y}')\;d\phi'
\end{align}
\begin{multline}
\int_0^{2\pi}\sin(\phi')e^{jk\rho'\sin\theta\cos(\phi-\phi')}\hat{\rho}'\;d\phi'
=\int_0^{2\pi}\sin(\phi')\cos(\phi')e^{jk\rho'\sin\theta cos(\phi-\phi')}\;d\phi'\hat{x'}\\
+\int_0^{2\pi}\sin^2(\phi')e^{jk\rho'\sin\theta\cos(\phi-\phi')}\;d\phi'\hat{y'}
\end{multline}
\begin{multline}
\int_0^{2\pi}\sin(\phi')\cos(\phi')e^{jk\rho'\sin\theta\cos(\phi-\phi')}\;d\phi'\hat{x'}=\\
\int_0^{2\pi}\sin(\phi')\cos(\phi')\cos({k\rho'\sin\theta\cos(\phi-\phi')}\;d\phi')\hat{x'}\\
+j\int_0^{2\pi}\sin(\phi')\cos(\phi')\sin({k\rho'\sin\theta\cos(\phi-\phi')}\;d\phi'\hat{x'}
\end{multline}
\begin{multline}
\int_0^{2\pi}\sin^2(\phi')e^{jk\rho'\sin\theta\cos(\phi-\phi')}\;d\phi'\hat{y'}=\\
\int_0^{2\pi}\sin^2(\phi')\cos({k\rho'\sin\theta\cos(\phi-\phi')})\;d\phi'\hat{y'}\\
+j\int_0^{2\pi}\sin^2(\phi')\sin({k\rho'\sin\theta\cos(\phi-\phi')})\;d\phi'\hat{y'}
\end{multline}
It was found through experimentation that the imaginary parts of both integrals are equal to zero.
\subsubsection{$\phi=0$}
\begin{multline}
\int_0^{2\pi}\sin(\phi')\cos(\phi')\cos({k\rho'\sin\theta\cos(\phi')}\;d\phi'\hat{x'}=0\\
\end{multline}
\begin{multline}
\int_0^{2\pi}\sin^2(\phi')\cos({k\rho'\sin\theta\cos(\phi')}\;d\phi'\hat{y'}\\
\end{multline}
\subsubsection{$\phi=\frac{\pi}{2}$}
\begin{multline}
\int_0^{2\pi}\sin(\phi')\cos(\phi')\cos({k\rho'\sin\theta\cos(\frac{\pi}{2}-\phi'))}\;d\phi'\hat{x'}\\
=\int_0^{2\pi}\sin(\phi')\cos(\phi')\cos({k\rho'\sin\theta\cos(\phi'))}\;d\phi'\hat{x'}
\end{multline}
\begin{multline}
\int_0^{2\pi}\sin^2\phi'\cos({k\rho'\sin\theta\cos(\frac{\pi}{2}-\phi')})\;d\phi'\hat{y'}\\
=\int_0^{2\pi}\sin^2\phi'\cos({k\rho'\sin\theta\sin\phi'})\;d\phi'\hat{y'}
\end{multline}
